CONSTRUCTION OF THE CHARACTERISTICS OF THE ASYNCHRONOUS MODE OF SYNCHRONOUS COMPENSATORS

NEBOLYUBOV Yu. E.

Associate Professor, Candidate of Technical Sciences

The need to investigate the asynchronous mode arose in connection with the author’s work on a relay-protection scheme for synchronous compensators against falling out of synchronism. For the proper selection and adjustment of this protection, it was necessary to consider not only the physical picture of the asynchronous mode, but also to derive certain quantitative relationships. Quantitative relations for the asynchronous mode of compensators can be obtained by constructing asynchronous characteristics, which are dependencies of the currents in the rotor and stator and of other quantities on the slip.

It is known that the asynchronous mode caused by loss of excitation may be a consequence of improper operation of the field-suppression automatic device or, in rare cases, a consequence of an open circuit in the excitation circuit.

The asynchronous mode of any synchronous machine proceeds considerably more complexly than that of an asynchronous machine. This is explained by certain features of synchronous machines: asymmetry of the rotor, different magnetic resistance along the direct and quadrature axes, and the absence of an excitation winding along the quadrature axis.

All modern methods for investigating the asynchronous mode of synchronous machines (Park, Davidson, Leneville) are based on replacing the actual synchronous machine by an equivalent circuit of a multi-winding transformer, on introducing the theory of two reactions and the method of symmetrical components.

Considerable accuracy and clarity in constructing asynchronous characteristics are provided by the method based on Park’s theory, who derived the fundamental equations of the excitation circuit and of torques in operational form and in per-unit quantities.

In Park’s equations, for the first time the interaction of the stator and rotor windings and of the rotor windings (excitation and damper windings) with one another is accurately taken into account.

The use of the method based on Park’s theory for constructing the asynchronous characteristics of a compensator can be divided into two principal stages: first, in general form for the transient mode, operational equations are derived for the currents in the rotor and stator [1]; second, these general equations are transformed in accordance with the features introduced by the asynchronous mode. From these expressions the asynchronous characteristics of the compensator are then constructed.

The construction of asynchronous characteristics of synchronous machines by the widely used Leneville method [4] takes, according to approximate estimates, three to four times more time and is associated with very cumbersome calculations.

The author has not encountered references in the literature to the possibility of constructing asynchronous characteristics—“current—slip” and “torque—slip”—by the method used in the present work.

Let us proceed to the determination of the basic relations underlying the method considered.

Current in the field winding (general operational equation)

The voltage applied to the field winding of a machine with damper circuits, according to Park, is balanced in the transient state by the voltage drops in the rotor circuits

\[ E_{fd}=I_{fd} r_{fd}+p x_{ffd} I_{fd}+p x_{f1d} I_{1d}-p x_{afd} i_d . \tag{1} \]

For the damper circuit, taking into account that the external voltage applied to it is equal to zero, we have

\[ 0=I_{1d} r_{1d}+p x_{f1d} I_{fd}+p x_{11d} I_{1d}-p x_{a1d} i_d . \tag{2} \]

In the equations presented, in per unit:

\[ \begin{aligned} I_{fd} &\quad \text{— current in the field winding;}\\ I_{1d} &\quad \text{— current in the damper winding;}\\ r_{fd} &\quad \text{— active resistance of the field winding along the longitudinal axis;}\\ r_{1d} &\quad \text{— active resistance of the equivalent damper circuit along the longitudinal axis;}\\ x_{f1d}=x_{1fd} &\quad \text{— mutual-induction reactance of the damper winding along the longitudinal axis and the field winding;}\\ x_{ffd} &\quad \text{— total reactance of the field winding along the longitudinal axis;}\\ x_{11d} &\quad \text{— total reactance of the damper winding along the longitudinal axis;}\\ x_{afd} &\quad \text{— mutual-induction reactance of the stator winding and the field winding;}\\ x_{a1d} &\quad \text{— mutual-induction reactance of the stator winding and the damper winding along the longitudinal axis;}\\ p=\dfrac{d}{dt} &\quad \text{— Heaviside operator;}\\ i_d &\quad \text{— longitudinal component of the stator current.} \end{aligned} \]

The simultaneous solution of equations (1) and (2) gives the values of the currents in the field winding and in the damper winding

\[ I_{fd}= \frac{(p x_{11d}+r_{1d})E_{fd}+\left[p^2(x_{11d}x_{afd}-x_{a1d}x_{f1d})+p x_{afd}r_{1d}\right]i_d} {p^2(x_{11d}x_{ffd}-x_{f1d}^{2})+p(x_{11d}r_{fd}+x_{ffd}r_{1d})+r_{1d}r_{fd}}; \tag{3} \]

\[ I_{1d}= \frac{-p x_{1fd}E_{fd}+\left[p^2(x_{ffd}x_{a1d}-x_{f1d}x_{afd})+p x_{a1d}r_{fd}\right]i_d} {p^2(x_{11d}x_{ffd}-x_{f1d}^{2})+p(x_{11d}r_{fd}+x_{ffd}r_{1d})+r_{1d}r_{fd}} . \tag{4} \]

Current in the field winding in the asynchronous mode of the compensator

The asynchronous mode of the compensator, caused by incorrect operation of the field-suppression automatic device or by an open circuit in the excitation circuit, is characterized by the fact that \(E_{fd}=0\). At the same time, it is necessary to introduce into equations (3) and (4) the slip \(s\) and to replace the operator \(p\).

It is known that the unit vectors of the stator-winding voltage along the direct and quadrature axes, \(e_d\) and \(e_q\), corresponding to the currents \(i_d\) and \(i_q\), can be expressed as follows: \(e_d=\sin\delta\); \(e_q=\cos\delta\), where \(\delta\) is the angle between the unit vector \(e\) and the vector \(e_q\).

For the asynchronous mode,

\[ \delta=\frac{\pi}{2}-st. \]

Therefore,

\[ e_d=\sin\left(\frac{\pi}{2}-st\right)=\cos st; \]

\[ e_q=\cos\left(\frac{\pi}{2}-st\right)=\sin st. \]

Let us introduce a system of vectors rotating with angular velocity \(s\) in relative units.

The voltage components along the axes \(e_d\) and \(e_q\) are the projections of two vectors \(e_{dm}\) and \(e_{qm}\) onto the time axis, rotating with velocity

In this case,

\[ e_d=1; \]

\[ e_q=-j; \]

\[ p=js. \]

The last expression \(p=js\) is obtained by double differentiation of the equation \(e\sin st=f(p,i)\).

After the first differentiation we have:

\[ se\cos st=pf(p,i) \]

and the second differentiation gives

\[ -s^2 e\sin st=p^2 f(p,i) \]

or

\[ p^2=-s^2, \]

whence

\[ p=js. \]

Putting \(E_{fd}=0\) and \(p=js\), from equations (3) and (4) we obtain the values of the currents in the field winding and damper winding for the asynchronous mode:

\[ I_{fd}= \frac{\left[-s^2\left(x_{11d}x_{afd}-x_{a1d}x_{f1d}\right)+jsx_{afd}r_{1d}\right]i_d} {-s^2\left(x_{11d}x_{ffd}-x_{f1d}^{2}\right)+js\left(x_{11d}r_{fd}+x_{ffd}r_{1d}\right)+r_{1d}r_{fd}}; \tag{5} \]

\[ I_{1d}= \frac{\left[-s^2\left(x_{ffd}x_{a1d}-x_{f1d}x_{afd}\right)+jsx_{a1d}r_{fd}\right]i_d} {-s^2\left(x_{11d}x_{ffd}-x_{f1d}^{2}\right)+js\left(x_{11d}r_{fd}+x_{ffd}r_{1d}\right)+r_{1d}r_{fd}}. \tag{6} \]

Substituting into equations (5) and (6) various slips \(s\) and the corresponding values of \(i_d\), one can determine a series of values of \(I_{fd}\) and \(I_{1d}\) and, from them, construct the asynchronous “current—slip” characteristics for the field winding and the damper winding.

The considerations underlying the determination of the stator currents along the direct axis \(i_d\) and along the quadrature axis \(i_q\), which are also used in constructing the “torque—slip” characteristic, are given below.

Current in the stator winding along the direct \((i_d)\) and quadrature \((i_q)\) axes of the compensator poles (general operational equations)

The starting point is the equations for the phase voltages:

\[ e_a = p\psi_a - r_a i_a; \tag{7} \]

\[ e_b = p\psi_b - r_b i_b; \tag{8} \]

\[ e_c = p\psi_c - r_c i_c. \tag{9} \]

Here \(i_a, i_b, i_c\) are the phase currents of the stator;

\(r_a, r_b, r_c\) are the active resistances of the stator phases;

\[ p = \frac{d}{dt} \]

is the Heaviside operator,

\(\psi_a, \psi_b, \psi_c\) are the flux linkages per phase.

The flux linkages of each phase may be expressed as follows:

\[ \psi_a = I_d\cos\Theta - I_q\sin\Theta - x_0 \frac{(i_a+i_b+i_c)}{3} - \frac{(x_d+x_q)}{3} \left( i_a-\frac{i_b+i_c}{2} \right) - \]

\[ - \frac{x_d-x_q}{3} \left[ i_a\cos 2\Theta - i_b\cos(2\Theta-120^\circ) + i_c\cos(2\Theta+120^\circ) \right]. \tag{10} \]

Similarly, for the other two phases:

\[ \psi_b = I_d\cos(\Theta-120^\circ) - I_q\sin(\Theta-120^\circ) - x_0 \frac{i_a+i_b+i_c}{3} - \frac{x_d+x_q}{3} \left( i_b-\frac{i_a+i_c}{2} \right) - \]

\[ - \frac{x_d-x_q}{3} \left[ i_a\cos(2\Theta-120^\circ) + i_b\cos(2\Theta+120^\circ) + i_c\cos 2\Theta \right]; \tag{11} \]

\[ \psi_c = I_d\cos(\Theta+120^\circ) - I_q\sin(\Theta+120^\circ) - x_0 \frac{i_a+i_b+i_c}{3} - \]

\[ - \frac{x_d+x_q}{3} \left( i_c-\frac{i_a+i_b}{2} \right) - \frac{x_d-x_q}{3} \left[ i_a\cos(2\Theta+120^\circ) + \right. \]

\[ \left. + i_b\cos 2\Theta + i_c\cos(2\Theta-120^\circ) \right], \tag{12} \]

where \(I_d\) and \(I_q\) are the excitation current in per-unit values along the direct and quadrature axes, respectively;

\(i_a, i_b, i_c\) are the phase currents of the stator;

\(x_d\) is the synchronous reactance of the stator winding along the direct axis;

\(x_0\) is the zero-sequence reactance of the stator winding;

\(\Theta\) is the angle between the direct axis and the axis of the first phase of the stator.

On the basis of the drawing given above (Fig. 1), the currents in the stator phases can be related to the current components along the direct and quadrature axes:

\[ i_d = \frac{2}{3} \left[ i_a\cos\Theta + i_b\cos(\Theta-120^\circ) + i_c\cos(\Theta+120^\circ) \right]; \]

\[ i_q = \frac{2}{3} \left[ i_a\sin\Theta + i_b\sin(\Theta-120^\circ) + i_c\sin(\Theta+120^\circ) \right]. \]

Using, further, the equations for the flux linkages of the stator system along the longitudinal and transverse axes:

\[ \psi_d = G_d(p)\bar E_d - x_d(p)i_d; \]

\[ \psi_q = G_q(p)\bar E_q - x_q(p)i_q \]

and the expressions for \(\psi_d\) and \(\psi_q\) as functions of the currents in the stator \(i_d\) and \(i_q\) and in the rotor \(I_d\) and \(I_q\):

\[ \psi_d = I_d - x_d i_d \quad \text{and} \quad \psi_q = I_q - x_q i_q, \]

we obtain equations for the currents in the excitation circuit:

\[ I_d = G_d(p)\bar E_d + [x_d - x_d(p)]i_d; \]

\[ I_q = G_q(p)\bar E_q + [x_q - x_q(p)]i_q, \]

where \(G_d(p), G_q(p), x_d(p), x_q(p)\) are the operational conductances and resistances along both axes;
\(\bar E_d\) and \(\bar E_q\) are the excitation voltages.

Fig. 1

To simplify the solution of these equations, Park proposed eliminating from them the phase quantities \(e_a, e_b, e_c, \psi_a, \psi_b, \psi_c\) and \(i_a, i_b, i_c\) by transition to the following auxiliary quantities along the \(d\) and \(q\) axes and to the zero-sequence phase quantities:

\[ e_d = \frac{2}{3}\left[ e_a\cos\Theta + e_b\cos(\Theta - 120^\circ) + e_c\cos(\Theta + 120^\circ) \right]; \]

\[ e_q = \frac{2}{3}\left[ e_a\sin\Theta + e_b\sin(\Theta - 120^\circ) + e_c\sin(\Theta + 120^\circ) \right]; \]

\[ e_0 = \frac{1}{3}(e_a + e_b + e_c); \]

\[ \psi_d = \frac{2}{3}\left[ \psi_a\cos\Theta + \psi_b\cos(\Theta - 120^\circ) + \psi_c\cos(\Theta + 120^\circ) \right]; \]

\[ \psi_q = \frac{2}{3}\left[ \psi_a\sin\Theta + \psi_b\sin(\Theta - 120^\circ) + \psi_c\sin(\Theta + 120^\circ) \right]; \]

\[ \psi_0=-\frac{1}{3}(\psi_a+\psi_b+\psi_c); \]

\[ i_0=\frac{1}{3}(i_a+i_q+i_c). \]

If, in the expressions for \(e_q\) and \(e_d\), the values \(e_a, e_b, e_c\) from the corresponding equations are substituted, and the derivatives \(p\psi_d\) and \(p\psi_q\) are formed, then one can find:

\[ e_d=p\psi_d-r_a i_d-\psi_q p\Theta; \tag{13} \]

\[ e_q=p\psi_q-r_a i_q-\psi_d p\Theta; \tag{14} \]

\[ e_0=p\psi_0-r_a i_0. \tag{15} \]

By an analogous substitution of the values of the flux linkages according to formulas (10), (11), and (12) into the expressions \(\psi_d, \psi_q, \psi_0\), we obtain:

\[ \psi_d=I_d-x_d i_d=G_d(p)E_d-x_d(p)i_d; \tag{16} \]

\[ \psi_q=I_q-x_q i_q=G_q(p)E_q-x_q(p)i_q; \tag{17} \]

\[ \psi_0=-x_0 i_0. \tag{18} \]

The expressions obtained for the voltages of the stator winding correspond to the steady-state condition under synchronism.

For the purposes of the present work, the expressions for the currents \(i_d\) and \(i_q\) as functions of \(e_d\) and \(e_q\), at constant rotor speed with allowance for a certain slip \(s\), are of interest. In the asynchronous regime, the currents and flux linkages are not constant quantities.

Expression of the stator currents \(i_d\) and \(i_q\) for the asynchronous regime

Taking into account that the asynchronous regime of the compensator is characterized by slip \(s\) and by the quantity \(p\Theta=1-s\), the equations for \(e_d\) and \(e_q\) (13 and 14) may be written as follows:

\[ e_d=p\psi_d-ri_d-(1-s)\psi_q; \tag{19} \]

\[ e_q=p\psi_q-ri_q-(1-s)\psi_d. \tag{20} \]

For the flux linkages \(\psi_d\) and \(\psi_q\) we obtain the expressions:

\[ \psi_d=G(p)E_d-x_d(p)i_d; \tag{21} \]

\[ \psi_q=-x_q(p)i_q. \tag{22} \]

For the convenience of algebraic transformations, introduce the auxiliary expressions:

\[ Z_d(p)=px_d(p)+r; \]

\[ Z_q(p)=px_q(p)+r. \]

Then the equations for \(e_d\) and \(e_q\) take the form:

\[ e_d=pG(p)E_d-Z_d(p)i_d+(1-s)x_q(p)i_q; \]

\[ e_q=(1-s)[G(p)E_d-x_d(p)i_d]-Z_q(p)i_q. \]

Solving these equations with respect to the currents \(i_d\) and \(i_q\), we obtain:

\[ i_d= \frac{ \left[pZ_q(p)+(1-s)^2x_q(p)\right]G(p)E_d -Z_q(p)e_d-(1-s)x_q(p)e_q }{ Z_d(p)Z_q(p)+(1-s)^2x_d(p)x_q(p) }; \tag{23} \]

\[ i_q= \frac{ (1-s)r\cdot G(p)E_d-Z_d(p)e_q+(1-s)x_d(p)e_d }{ Z_d(p)Z_q(p)+(1-s)^2x_d(p)x_q(p) }. \tag{24} \]

As was indicated earlier, in the asynchronous mode:

\[ E_d=0;\quad e_d=\cos st;\quad e_q=\sin st. \]

Introducing a system of rotating vectors with angular velocity \(s\) and replacing \(p=js\), the currents \(i_d\) and \(i_q\) may be expressed in complex form:

\[ i_d= \frac{jsx_q(js)+r-j(1-s)x_q(js)} {[jsx_d(js)+r][jsx_q(js)+r]+(1-s)^2x_d(js)x_d(js)x_q(js)} . \]

or

\[ i_d= \frac{jx_q(js)-\dfrac{r}{1-2s}} {x_d(js)x_q(js)+\dfrac{r}{1-2s}\left\{r+js[x_d(js)+x_q(js)]\right\}}; \tag{25} \]

\[ i_q= \frac{x_d(js)+\dfrac{jr}{1-2s}} {x_d(js)x_q(js)+\dfrac{r}{1-2s}\left\{r+js[x_d(js)+x_q(js)]\right\}} . \tag{26} \]

Let us proceed to the determination of the operational reactive resistances \(x_d(p)\) and \(x_q(p)\) in general form.

Operational reactive resistances along the direct \(x_d(p)\) and quadrature \(x_q(p)\) pole axes

(General expressions)

The flux linkages of the stator winding along the direct pole axis may be represented in the form

\[ \psi_d=x_{a1d}i_d+x_{a1d}i_{1d}-x_di_d . \]

Let us replace the currents \(i_d\) and \(i_{1d}\) according to equations (3) and (4). Then:

\[ \psi_d= \frac{p(x_{11d}x_{afd}-x_{f1d})+x_{afd}r_{1d}}{A(p)}E_{fd} - \left[ x_d- \frac{p^2(x_{11d}x_{afd}-x_{f1d}x_{a1d}^{2}x_{afd}+x_{ffd}^{2}x_{a1d})}{A(p)} + \frac{p(x_{afd}^{2}r_{1d}+x_{a1d}^{2}r_{fd})}{A(p)} \right] i_d . \]

Analogously, for the quadrature pole axis, the flux linkages of the stator winding are expressed as

\[ \psi_q=- \left[ x_q-\frac{px_{a1q}}{px_{11q}+r_{1q}} \right]i_q . \]

Consequently, in the presence of one damper circuit on the rotor we have

\[ G(p)= \frac{p(x_{11d}x_{a1d}-x_{f1d}x_{a1d})+x_{afd}r_{1d}} {A(p)} \]

where

\[ A(p)=p^2(x_{11d}x_{ffd}-x_{f1d}^2)+p(x_{11d}r_{fd}+x_{ffd}r_{1d})+r_{1d}r_{fd}; \]

\[ x_d(p)=x_d-\left[ \frac{p^2(x_{11d}x_{afd}^2-2x_{f1d}x_{a1d}x_{afd}+x_{ffd}^2x_{a1d})}{A(p)} + \frac{p(x_{afd}^2r_{1d}+x_{a1d}^2r_{fd})}{A(p)} \right]; \tag{27} \]

\[ x_q(p)=x_q-\frac{px_{a1q}}{px_{11q}+r_{1q}}. \tag{28} \]

Resistances \(x_d(p)\) and \(x_q(p)\) in the asynchronous mode

Substituting \(p=js\) into equations (27) and (28), we obtain the relations for the asynchronous mode:

\[ x_d(js)=x_d-\frac{-s^2(x_{11d}x_{afd}^2-2x_{f1d}x_{a1d}x_{afd}+x_{ffd}^2x_{a1d})}{A(js)} + \frac{(js)(x_{afd}^2r_{1d}+x_{a1d}^2r_{fd})}{A(js)}; \tag{29} \]

\[ x_q(js)=x_q-\frac{(js)x_{a1q}}{(js)x_{11q}+r_{1q}}. \tag{30} \]

In equation (29)

\[ A(js)=-s^2(x_{11d}x_{ffd}-x_{f1d}^2)+(js)(x_{11d}r_{fd}+x_{ffd}r_{1d})+r_{1d}r_{fd}. \]

Torque of the compensator in the asynchronous mode

(General expression)

First of all it is necessary to determine the torque in the steady-state mode \((s=0)\).

The power delivered by the stator of any synchronous machine is

\[ P=M_p\Theta+P_1-P_2, \]

where \(M_p\Theta\) is the mechanical power transmitted through the air gap (\(M\) is the torque);

\(P_1\) is the instantaneous power due to the reserve of magnetic energy;

\[ P_2=r(i_d^2+i_q^2+i_0^2) \]

is the total loss in the active resistance of the stator winding.

From this equation the expression for the torque is derived.

At the same time, the torque depends on the currents in the stator and rotor circuits: \(i_d, i_q, i_0, I_d, I_q\), the values of which were given earlier.

If the operation of the compensator is considered with values \(i_d, i_q, i_0, I_d, I_q\) remaining unchanged for a certain interval of time, then the magnetic energy stored in the machine does not change during rotation of the rotor. Therefore the power delivered by the rotor is equal to, and opposite in sign to, the losses in the stator.

Then the power equation is

\[ P=M_p\Theta-r(i_d^2+i_q^2+i_0^2). \tag{31} \]

If we express the losses in the stator copper through the phase currents, then we obtain

\[ P = M p \Theta - \frac{2}{3} r \left(i_a^2 + i_b^2 + i_c^2\right). \tag{32} \]

Here the relation has been used

\[ \frac{2}{3}\left(i_a^2 + i_b^2 + i_c^2\right) = i_d^2 + i_q^2 + i_0^2, \]

obtained on the basis of Park’s general equations.

On the other hand, the magnetic power of a synchronous machine is proportional to the sum of the products of current and voltage in each phase,

\[ P = \frac{2}{3}\left(e_a i_a + e_b i_b + e_c i_c\right). \]

Using the relation between the phase currents and voltages and the corresponding values along the direct and quadrature axes, we obtain

\[ P = e_d i_d + e_q i_q + e_0 i_0. \tag{33} \]

From equations (31) and (32) it follows that

\[ M = \frac{e_d i_d + e_q i_q + e_0 i_0 + r\left(i_d^2 + i_q^2 + i_0^2\right)}{p\Theta}. \tag{34} \]

For the operating mode of the machine under consideration, with currents in all circuits unchanged for a certain instant, the equations for \(e_d\), \(e_q\), and \(e_0\) (13–15) take the form:

\[ e_d = -\psi_q p\Theta - r i_d; \]

\[ e_q = \psi_d p\Theta - r i_q; \]

\[ e_0 = -r i_0. \]

Substituting these expressions into the torque equation (34), after a series of transformations we obtain:

\[ M = i_q \psi_d - i_d \psi_q. \tag{35} \]

The expression obtained for the torque corresponds, as was indicated, to the steady-state condition under synchronism.

The expression for the rotational torque in the asynchronous mode can be obtained if the currents \(i_d\) and \(i_q\) are replaced by their values in the asynchronous mode according to equations (25) and (26).

Expression for the rotational torque of the compensator in the asynchronous mode

Substituting the currents \(i_d\) and \(i_q\) into the power equation, we obtain

\[ P = \frac{1}{2}\left[1 \cdot i_d - j i_q\right] = \frac{1}{2}\left(\text{real part } i_d - \text{imaginary part } i_q\right). \]

Before determining the average rotational torque \(M_{\mathrm{av}}\), it is necessary to find expressions for the flux linkages \(\psi_d\) and \(\psi_q\).

Basic equations in the asynchronous mode:

\[ e_d + r i_d = p\psi_d - (1-s)\psi_q; \]

\[ e_q + r i_q = (1-s)\psi_d + p\psi_q. \]

Whence we find (for \(p=js\)):

\[ \psi_d=\frac{js(e_d+r i_d)+(1-s)(e_q+r i_q)}{(1-2s)}; \]

\[ \psi_q=\frac{js(e_q+r i_q)-(1-s)(e_d+r i_d)}{(1-2s)}. \]

Since \(e_d=1\) and \(e_q=-j\), then

\[ \psi_d=-j+\frac{r}{1-2s}\left[jsi_d+(1-s)i_q\right]; \]

\[ \psi_q=-1+\frac{r}{1-2s}\left[jsi_q-(1-s)i_d\right]. \]

The scalar product of the vectors \(\overline{\Psi}_d\) and \(\overline{i}_q\); \(\overline{\Psi}_q\) and \(\overline{i}_d\) gives the expression for the mean torque

\[ M_{\mathrm{av}}=\frac{1}{2}\left[i_q\Psi_d-i_d\Psi_q\right]= \]

\[ =\left\{ i_q(-j)+i_q\frac{r}{1-2s}\left[jsi_d+(1-s)i_q\right]- \right. \]

\[ \left. -i_d(-1)-i_d\frac{r}{1-2s}\left[jsi_q-(1-s)i_d\right] \right\}; \]

\[ M_{\mathrm{av}}=P_{\mathrm{av}}+r\frac{i_d^2+i_q^2}{2}+\frac{rs}{2(1-s)}(i_q-ji_d)^2. \tag{36} \]

For the current components of direct and reverse rotation we have

\[ i_A=\frac{1}{2}(i_d+ji_q); \tag{37} \]

\[ i_B=\frac{1}{2}(i_d-ji_q) \tag{38} \]

and analogously for the corresponding voltages

\[ e_A=\frac{1}{2}(e_d+je_q); \]

\[ e_A=\frac{1}{2}(e_d-je_q). \]

Substituting into the expressions for \(i_A\) and \(i_B\) the values of the currents \(i_d\) and \(i_q\) obtained earlier, we find:

\[ i_A=\frac{1}{2} \frac{ \dfrac{-2r}{1-2s}+j\left[x_d(js)+x_q(js)\right] }{ x_d(js)x_q(js)+\dfrac{r}{1-2s}\left\{r+js\left[x_d(js)+x_q(js)\right]\right\} }; \]

\[ i_B=\frac{1}{2} \frac{ j\left[x_d(js)-x_q(js)\right] }{ x_d(js)x_q(js)+\dfrac{r}{1-2s}\left\{r+js\left[x_q(js)+x_q(js)\right]\right\} }. \]

According to Park’s theory, the average torque in the asynchronous mode with loss of excitation, on the basis of equation (36), will yield the following expression:

\[ M_{\text{st}}=\frac{1}{2}\left[\text{real part } i_d-\text{imaginary part } i_q\right]+r i_A^2+\frac{r}{1-2s}i_B^2 . \]

From this equation the “torque—slip” characteristic of the asynchronous mode of the compensator is constructed.

If the speed reaches the semisynchronous value, then the reverse-rotation current \(i_B\) becomes equal to zero, and a sharp dip appears in the torque curve near the semisynchronous speed (the current \(i_B\) enters the torque equation in the form of the term \(\dfrac{r}{1-2s} i_B^2\)).

The total stator current, obtained by superposing the current \(i_A\) of the fundamental frequency on the current \(i_B\) of frequency \((1-2s)\), pulsates incorrectly in time, because at speeds differing from zero, one-half, and full synchronous speed there is a reverse-rotation current \(i_B\).

The maximum value of the pulsating current \(\overline{i}_a=\left|\overline{i}_A+\overline{i}_B\right|\) is obtained when the phase axis \(a\) and the direct axis coincide, and the minimum value is

\[ \overline{i}_a=\left|\overline{i}_A-\overline{i}_B\right|. \]

Considering the obtained relations, it can be seen that the current in the field winding \(I_{fd}\) and the current in the damper winding \(I_{1d}\) are functions of the same machine parameters that were used to determine the stator currents \(i_d\) and \(i_q\) and the impedances \(x_d(js)\) and \(x_q(js)\).

Comparing the equations, one may note that the denominators in the expressions for \(I_{fd}\) and \(I_{1d}\), as well as in the equations for the impedances \(x_d(js)\) and \(x_q(js)\), are identical.

This circumstance considerably simplifies the calculations given below in the calculation example for the synchronous compensator of the Elektrosila plant.

Procedure for calculation by the method based on Park’s theory

1. The impedances \(x'_d(js)\) and \(x_q(js)\) are determined at various slips.

For this purpose, a complete calculation of the magnetic circuit of the synchronous compensator is carried out and then, by the formulas of Linville (1) and Alger [5], all quantities entering into the expressions for these impedances are found.

1. From the known impedances \(x_d(js)\) and \(x_q(js)\), in complex form, the components of the stator current \(i_d\) and \(i_q\) are found, likewise in complex expression.

2. The currents \(i_d\) and \(i_q\) are used in the equations giving the value of the current in the field winding \(I_{fd}\) at various slips.

3. The currents \(i_A\) and \(i_B\) are determined, and from them the values of the torque in the asynchronous mode are obtained, likewise for various slips \((M_{\text{av}})\).

In the specific determination of \(x_d(js)\), \(x_q(js)\), \(i_d\), \(i_q\), \(I_{fd}\), \(i_A\), \(i_B\), \(M_{\text{av}}\), all expressions are considerably simplified, and all operations reduce to the transformation of complex dependences.

Figure 2

Visible plot annotations: \(x_d, x'_q\); \(x_d(jS)=f(S)\); \(x'_q(jS)=f(S)\); \(x_d(jS)=r+jr'\); \(x'_q(jS)=g+jg'\); \(S\).

Figure 3

Visible plot annotations: \(l_d\); \(l_d=f(S)\); \([l_d]\); \(l_d=b+jb'\); \(S\).

Figure 4

Visible plot annotations: \(l_q\); \(l_q=f(S)\); \([l_q]\); \(l_q=c+jc'\); \(S\).

Figure 5

Visible plot annotations: \(l_{rd}\); \(l_{rd}=f(S)\); \([l_{rd}]\); \(l_{rd}=a+ja'\); \(S\).

Example of constructing the asynchronous characteristics of the WR-446-750 synchronous compensator of the “Elektrosila” plant

All the parameters necessary for the calculation are given in Table 1.

Table 1

To construct the asynchronous characteristics of the compensator, “current—slip” and “torque—slip,” the resistances \(x_d(js)\) and \(x_q(js)\) are first found for various slips. Then from them the currents \(i_d\) and \(i_q\), and the current \(I_{fd}\), are determined, and the “current—slip” curves are plotted.

From the values of the currents \(i_d\) and \(i_q\), the currents of forward rotation \((i_A)\) and reverse rotation \((i_B)\) are calculated, after which the “torque—slip” characteristic can be constructed.

An example of the calculation is given only for one point \(S=1\) and is shown in Table 2. The remaining points of the characteristics are calculated similarly for different values of the slip \(S\).

Table 2

Similarly,

\[ x_q(jS)=0.611-\frac{jS\cdot 0.335}{jS\cdot 0.668+0.0397}. \]

Substituting various values of slip into the last expressions \(x_d(jS)\) and \(x_q(jS)\), we find the quantities necessary for constructing the curves of the currents \(i_d\) and \(i_q\).

For \(S=1\) we obtain:

\[ x_d(jS)=a+ja'=0.111-j\,0.0314; \]

\[ x_q(jS)=b+jb'=0.111-j\,0.03. \]

Components of the stator current along the direct axis \((i_d)\) and along the quadrature axis \((i_q)\) for \(S=1\)

According to formula (25) we have

for \(S=1\)

\[ i_d= \frac{ j(0.111-j\,0.03)-\dfrac{0.0012}{1-2} }{ (0.111-j\,0.0314)(0.111-j\,0.03)+\dfrac{0.0042}{1-2} } \cdot \frac{1}{ \bigl[0.0042+j(0.111-j\,0.0314+0.111-j\,0.03)\bigr] } = \]

\[ =2.58+j\,6.77 \]

\[ i_d=C+jc''=-2.58+j\,6.77. \]

For the current \(i_q\), according to expression (26), we obtain

for \(S=1\)

\[ i_q= \frac{ (0.111-j\,0.0314)+\dfrac{j\,0.0042}{1-2} }{ (0.111-j\,0.0314)(0.111-j\,0.0314)+\dfrac{0.004}{1-4} } \cdot \frac{1}{ \bigl[0.0042+j(0.111-j\,0.0314+0.111-j\,0.03)\bigr] } = \]

\[ =8.22+j\,2.62. \]

\[ i_q=8.28+j\,2.62 \]

Current in the field winding \(I_{fd}\).

For \(S=1\).

According to equation (5), the current in the field winding is equal to

(after substitution of the data from Table 1):

\[ I_{fd}= \frac{ \bigl[-S^2(0.827\cdot 0.968\cdot 0.706\cdot 0.815)+jS\cdot 0.908\cdot 0.0534\bigr]\,i_d }{ -S^2(0.827\cdot 0.968-0.706^2)+jS(0.827\cdot 0.052+0.968\cdot } \]

\[ \cdot 0.0534)+0.0534\cdot 0.052 = \]

\[ = \frac{\bigl[-S^2\cdot 0.174+jS\cdot 0.0485\bigr]\,i_d} {-S^2\cdot 0.335+jS\cdot 0.097+0.0028}. \]

If in the last expression for \(I_{fd}\) we substitute various slips and the corresponding values of \(i_d\), then we can obtain the characteristic \(I_{fd}=f(S)\).

For \(S=1\):

\[ I_{fd}= \frac{(-0.174+j\,0.0485)(-2.58+j\,6.77)} {j\,0.097-0.322} = \]

\[ =-1.357+j\,3.434. \]

\[ I_{fd}=e+e'=-1.357+j\,3.434. \]

Currents of forward \((j_A)\) and reverse \((i_B)\) rotation.

For \(S=1\).

\[ i_A=\frac{1}{2}(-2.58+j\,6.77+j\,8.22-2.62) \]

\[ =-2.6+j\,7.495;\qquad |i_A=7.92| \]

\[ i_B=\frac{1}{2}(-2.58+j\,6.77-j\,8.22+2.62)= \]

\[ =0.02-j\,0.725;\qquad |i_B=0.725|. \]

The results of all calculations for constructing the characteristics “current—slip” and “torque—slip” are presented in Tables 3 and 4.

Figure 2 shows the variation of the reactances \(x_d(js)\) and \(x_q(js)\) as a function of slip.

The asynchronous characteristics “current—slip” are plotted in Figs. 3, 4, 5.

The curve \(I_{fd} = f(S)\) (Fig. 5) has a sharp dip near half-synchronous speed, a rapid rise, then a sharp fall, a steep rise, and a steep decrease to zero near synchronism. Obviously, the smaller

Fig. 6.

Fig. 7.

the poles, i.e., the greater the difference between a machine with salient poles and a machine with a smooth rotor, the greater the influence of asymmetry and the greater the dip in the field-winding current curve at half-synchronous speed.

An analogous variation is exhibited by the components of the stator currents along the axes \(i_d\) and \(i_q\) (Figs. 4 and 5).

In Figs. 6 and 7 the dependences of the currents of forward and reverse rotation \((i_A\) and \(i_B)\) and of the torque \(M\) on slip are plotted (Fig. 8).

Table 3

Data for constructing the “torque—slip” characteristic \((M=f(S))\)

Table 4

Data for constructing the “current—slip” characteristics \((i_d=f(S);\ i_q=f(S);\ I_{fd}=f(S))\)

The reverse-rotation current (Fig. 7) at semisynchronous speed is equal to zero. This causes a dip in the curve of the asynchronous torque. In our example the ratio of the compensator parameters is such that the dip

Fig. 8

at \(S = 0.5\) is considerable, but the possibility of the machine being braked is excluded because the resisting torque on the compensator shaft is very small.

LITERATURE

1. Lüter R. A. Theory of Transient Conditions of Synchronous Machines. Leningrad, 1939.

2. Park R. H. “Two Reactions Theory of Synchronous Machines.” P. II. Transactions AIEE, 1929, I. D., p. 717.

3. Park R. H. “Two Reactions Theory of Synchronous Machines.” P. I. Transactions AIEE, 1933 Dec., p. 353.

4. Linville T. M. “Srarting Performance of Salient Pole Synchronous Motors.” Trans. AIEE, 1930, I. A., p. 531.

5. Alger P. L. “The Calculation of Armature Reactance of Synchronous Machines.” Trans. AIEE, vol. 47, 1928.